A note on Two-Point Concentration of the Independence Number of $G_{n,m}$

We show that the independence number of $G_{n,m}$ is concentrated on two values for $n^{5/4+ \epsilon} < m \le \binom{n}{2}$. This result establishes a distinction between $G_{n,m}$ and $G_{n,p}$ with $p = m/ \binom{n}{2}$ in the regime $n^{5/4 + \epsilon} < m< n^{4/3}$. In this regime the independence number of $G_{n,m}$ is concentrated on two values while the independence number of $G_{n,p}$ is not; indeed, for $p$ in this regime variations in $\alpha( G_{n,p})$ are determined by variations in the number of edges in $G_{n,p}$.